Trigonometry MCQ1) Which of the following is the correct value of cot 10^{0}.cot 20^{0}.cot 60^{0}.cot 70^{0}.cot 80^{0}?
Answer: (a) 1/√3 Explanation: Here, we can apply the formula  cot A. cot B = 1 (when A + B = 90^{0}) = (cot 20^{0} . cot 70^{0}) x (cot 10^{0} . cot 80^{0}) x cot 60^{0} = 1 x 1 x 1/√3 = 1/√3 So, the correct value of cot 10^{0}.cot 20^{0}.cot 60^{0}.cot 70^{0}.cot 80^{0} = 1/√3 2) If a sin 45^{0} = b cosec 30^{0}, what is the value of a^{4}/b^{4}?
Answer: (b) 4^{3} Explanation: Given a sin 45^{0} = b cosec 30^{0} So, a/b = cosec 30^{0}/ sin 45^{0} a/b = 2/( 1/√2) a/b = 2√2/1 a^{4}/b^{4} = (2√2/1)^{4} a^{4}/b^{4} = 64/1 or, a^{4}/b^{4} = 4^{3} 3) If tan θ + cot θ = 2, then what is the value of tan^{100} θ + cot^{100} θ?
Answer: (c) 2 Explanation: Given tan θ + cot θ = 2 Put θ = 45^{0}, above equation will satisfy as, 1 + 1 = 2 So, θ = 45^{0}, = tan^{100} 45^{0} + cot^{100} 45^{0} = 1^{100} + 1^{100} = 2 4) If the value of α + β = 90^{0}, and α : β = 2 : 1, then what is the ratio of cos α to cos β ?
Answer: (c) 1 : √3 Explanation: Given α + β = 90^{0}, and α : β = 2 : 1 So, we can say that 2x + x = 90^{0} 3x = 90^{0}, which give x = 30^{0} So, α = 2x = 60 β = x = 30 cos α / cos β = cos 60^{0} / cos 30^{0} => (1/2) / (√3/2) or, 1/2 * 2/√3 = 1/√3 Or the ratio between cos α : cos β = 1 : √3 5) If θ is said to be an acute angle, and 7 sin2 θ + 3 cos2 θ = 4, then what is the value of tan θ?
Answer: (c) 1/√3 Explanation: Given 7 sin^{2} θ + 3 cos^{2} θ = 4 => 7 sin^{2} θ + 3 (1  sin^{2} θ) = 4 => 7 sin^{2} θ + 3  3sin^{2} θ = 4 Then, 4sin^{2} θ = 1 Or, sin θ = 1/2 So, θ = 30^{0} Now, put θ = 30^{0} in tan θ, we will get, tan θ = 1/√3 Alternate We can directly check the equation by putting values of θ. Let's put θ = 30^{0} 7 sin^{2} 30^{0} + 3 cos^{2} 30^{0}= 4 Then, 7 * 1/4 + 3 * 3/4 = 4 So, 7/4 + 9/4 = 4 16/4 = 4 Or, 4 = 4 (so, it satisfy the condition) Now, tan 30^{0} = 1/√3 6) If tan θ  cot θ = 0, what will be the value of sin θ + cos θ?
Answer: (b) √2 Explanation: Given tan θ  cot θ = 0 Let's put θ = 45^{0} in order to satisfy the above equation tan 45^{0}  cot 45^{0} = 0 1  1 = 0 (equation satisfied with θ = 45^{0}) Now, put θ = 45^{0} in sin θ + cos θ, we will get = sin 45^{0} + cos 45^{0} = 1/√2 + 1/√2 = √2 7) If θ is said to be an acute angle, and 4 cos^{2} θ  1 = 0, then what is the value of tan (θ  15^{0})?
Answer: (a) 1 Explanation: Given 4 cos^{2} θ  1 = 0 4 cos^{2} θ = 1 cos^{2} θ = 1/4 cos θ = 1/2 Or, θ = 60^{0} So, tan (θ  15^{0}) = ? => tan (60^{0}  15^{0}) = tan 45^{0} = 1 8) If the value of θ + φ = π/2, and sin θ = 1/2, what will be the value of sinφ?
Answer: (c) √3/2 Explanation: Given θ + φ = π/2 It can be written as, θ + φ = 90^{0} (as π = 180^{0}) …….(i) sin θ = 1/2 or, θ = 30^{0} On putting the value of θ = 30^{0} in equation (i), we will get, 30^{0} + φ = 90^{0} So, φ = 60^{0} Then, sin φ = sin 60^{0} = √3/2 9) What will be the value of 2cos^{2} θ  1, if cos^{4} θ  sin^{4} θ = 2/3?
Answer: (d) 2/3 Explanation: Given cos^{4} θ  sin^{4} θ = 2/3 Now, here we can apply the formula  a^{4}  b^{4} = (a^{2}  b^{2}) (a^{2} + b^{2}) So, (cos^{2} θ  sin^{2} θ) (cos^{2} θ + sin^{2} θ) = 2/3 So, 1 x (cos^{2} θ  sin^{2} θ) = 2/3 (because cos^{2} θ + sin^{2} θ = 1) => cos^{2} θ  (1  cos^{2} θ) = 2/3 (because sin^{2} θ = 1  cos^{2} θ) So, 2cos^{2} θ  1 = 2/3 10) What will be the value of 1  2sin^{2} θ, if cos^{4} θ  sin^{4} θ = 2/3?
Answer: (d) 2/3 Explanation: Given cos^{4} θ  sin^{4} θ = 2/3 Now, here we can apply the formula  a^{4}  b^{4} = (a^{2}  b^{2}) (a^{2} + b^{2}) So, (cos^{2} θ  sin^{2} θ) (cos^{2} θ + sin^{2} θ) = 2/3 So, 1 x (cos^{2} θ  sin^{2} θ) = 2/3 (because cos^{2} θ + sin^{2} θ = 1) => (1  sin^{2} θ)  sin^{2} θ = 2/3 So, 1  2sin^{2} θ = 2/3 11) What is the value of tan θ/(1  cot θ) + cot θ/(1  tan θ)?
Answer: (a) tan θ + cot θ + 1 Explanation: tan θ/(1  (1/tan θ) + (1/tan θ)/(1  tan θ) = tan^{2} θ/ (tan θ  1)  1/tan θ(tan θ  1) = tan^{3} θ  1/tan θ(tan θ  1) Apply the formula, a^{3} b^{3} = (a  b) (a^{2} + ab + b^{2}) = (tan θ  1) (tan^{2} θ + tan θ + 1) / tan θ(tan θ  1) = (tan^{2} θ + tan θ + 1) / tan θ On taking tan θ common from the numerator, we will get, = tan θ + cot θ + 1 12) What is the value of sin θ/(1 + cos θ) + sin θ/(1  cos θ), where (0^{0} < θ < 90^{0})?
Answer: (a) 2cosec θ Explanation: Given, sin θ/(1 + cos θ) + sin θ/(1  cos θ) = [sin θ (1  cos θ) + sin θ (1 + cos θ)] / [(1  cos θ) (1 + cos θ)] = [sin θ  sin θ cos θ + sin θ + sin θ cos θ] / [1  cos^{2} θ] = 2 sin θ / sin^{2} θ = 2 cosec θ 13) What will be the value of (√3 tanθ + 1), if r sinθ = 1, and r cosθ = √3?
Answer: (a) 2 Explanation: Given, r sinθ = 1, and r cosθ = √3 r sinθ / r cosθ = 1/√3 tanθ = 1/√3 or √3 tanθ = 1 So, √3 tanθ + 1= 1 + 1 = 2 14) What is the value of (tan^{2} θ  sec^{2} θ)?
Answer: (b) 1 Explanation: (tan^{2} θ  sec^{2} θ) = sin^{2} θ/cos^{2} θ  1/cos^{2} θ = (sin^{2} θ  1) / cos^{2} θ =  cos^{2} θ/cos^{2} θ = 1 15) If sin θ = 0.7, then what is the value of cosθ, if 0^{0} <= θ < 90^{0}?
Answer: (a) √0.51 Explanation: Given sin θ = 0.7 As we know, sin^{2} θ + cos ^{2} θ = 1 So, (0.7)^{2} + cos ^{2} θ = 1 Then, 0.49 + cos ^{2} θ = 1 => cos^{2} θ = 1  0.49 cos θ = √0.51 16) What is the value of tan3θ, If tan7θ.tan2θ = 1?
Answer: (b) 1/√3 Explanation: Given tan7θ.tan2θ = 1 As we know, if tanA . tanB = 1 then, A + B = 90^{0} So, 7θ + 3θ = 90^{0} => 9θ = 90^{0} Or, θ = 10^{0} Now, we have to find tan3θ So, put θ = 10^{0} in tan3θ, we will get tan 30^{0} = 1/√3 17) What will be the value of 3cos80^{0}.cosec10^{0} + 2cos59^{0}.cosec31^{0}?
Answer: (c) 5 Explanation: 3cos80^{0}.cosec10^{0} + 2cos59^{0}.cosec31^{0} = ? According to the identity, [if A + B = 90^{0} then, cosA.cosecB = 1] So, 3cos80^{0}.(1/sin10^{0}) + 2cos59^{0}.(1/sin31^{0}) = 3cos80^{0}.(1/sin(90^{0}  80^{0})) + 2cos59^{0}.(1/sin(90^{0}  59^{0})) => 3cos80^{0}/cos 80^{0}) + 2cos59^{0}/cos 59^{0} ( because sin (90^{0}  θ) = cos θ) = 3 + 2 = 5 18) If sin (θ + 18^{0}) = cos 60^{0}, then what is the value of cos5θ, where 0^{0} < θ < 90^{0}?
Answer: (b) 1/2 Explanation: Given sin (θ + 18^{0}) = cos 60^{0} sin (θ + 18^{0}) = cos (90^{0}  30^{0}) So, sin (θ + 18^{0}) = sin30^{0} Then, θ = 30^{0}  18^{0} θ = 12^{0} So, cos5θ = cos 5 x 12^{0} = cos 60^{0} = 1/2 19) If cos A = 2/3, then what is the value of tan A?
Answer: (d) √5/2 Explanation: According to the trigonometric identities, 1 + tan^{2} A = sec^{2} A And we know, sec A = 1/cos A So, sec A = 1/(2/3) = 3/2 Then, 1 + tan^{2} A = (3/2)^{2} = 9/4 => tan^{2} A = 9/4  1 => tan^{2} A = 5/4 So, tan A = √5/2 20) What will be the simplified value of (sec A sec B + tan A tan B)^{2}  ( sec A tan B + tan A sec B)^{2}?
Answer: (b) 1 Explanation: The question is in the form of (a + b)^{2} So, on applying the identity, and after expanding the given equation, we will get  => sec^{2} A sec^{2} B + tan^{2} A tan^{2} B + 2 sec A sec B tan A tan B  sec^{2} A tan^{2} B  tan^{2} A sec^{2} B  2 sec A tan B tan A sec B => Then, sec^{2} A [sec^{2} B  tan^{2} B]  tan^{2} A [sec^{2} B  tan^{2} B] So, it will be written as [sec^{2} A  tan^{2} A] [sec^{2} B  tan^{2} B] = 1 x 1 = 1. 21) What is the simplified value of (cosec A  sin A)^{2} + (sec A  cos A)^{2}  (cot A  tan A)^{2}?
Answer: (b) 1 Explanation: The question is in the form of (a  b)^{2} (a  b)^{2} = a^{2} + b^{2}  2ab So, on applying the identity, and after expanding the given equation, we will get  => cosec^{2} A + sin^{2} A  2 cosec A sin A + sec^{2} A + cos^{2} A  2 sec A cos A  cot^{2} A  tan^{2} A + 2 cot A tan A After solving it with using trigonometric identities, we will get  => (cosec^{2} A  cot^{2} A) + (sin^{2} A + cos^{2} A) + (sec^{2} A  tan^{2} A) 2 = 1 + 1 + 1  2 = 3  2 = 1 Alternate method (cosec A  sin A)^{2} + (sec A  cos A)^{2}  (cot A  tan A)^{2} We can solve it directly by putting θ = 45^{0} = (cosec 45^{0}  sin 45^{0})^{2} + (sec 45^{0}  cos 45^{0})^{2}  (cot 45^{0}  tan 45^{0})^{2} = (√2  1/√2)^{2} + (√2  1/√2)^{2}  (1  1)^{2} = 1/2 + 1/2  0 = 1 22) What will be the value of sec^{4} θ  tan^{4} θ, if sec^{2} θ + tan^{2} θ = 7/12?
Answer: (b) 7/12 Explanation: Given sec^{2} θ + tan^{2} θ = 7/12 Now, here we can apply the formula  a^{4}  b^{4} = (a^{2}  b^{2}) (a^{2} + b^{2}) sec^{4} θ  tan^{4} θ = (sec^{2} θ  tan^{2} θ) (sec^{2} θ + tan^{2} θ) = 1 x (sec^{2} θ + tan^{2} θ) {because 1 + tan^{2} θ = sec^{2} θ} = 1 x 7/12 = 7/12 23) What is the value of cos^{2} 20^{0} + cos^{2} 70^{0}?
Answer: (c) 1 Explanation: cos^{2} 20^{0} + cos^{2} 70^{0} We can write cos^{2} 20^{0} as cos^{2} (90^{0}  70^{0}) So, cos^{2} (90^{0}  70^{0}) + cos^{2} 70^{0} = sin^{2} 70^{0} + cos^{2} 70^{0} = 1 24) If r cosθ = √3, and r sinθ = 1, what is the value of r^{2} tanθ?
Answer: (a) 4/√3 Explanation: Given r cosθ = √3, and r sinθ = 1 r cosθ / r sinθ = 1/√3 tanθ = tan 30^{0} Or, θ = 30^{0} On putting, θ = 30^{0}, we will get, r sin 30^{0} = 1 r x ½ = 1 or r =2 Now, r^{2} tanθ = ? = (2)^{2} tan 30^{0} = 4 x 1/√3 = 4/√3 25) Which of the following is the correct value of tan^{2} A + cot^{2} A  sec^{2} A cosec^{2} A, where 0^{0} < A < 90^{0}?
Answer: (c) 2 Explanation: We can solve it by putting θ = 45^{0} On putting θ = 45^{0}, we will get  = tan^{2} 45^{0} + cot^{2} 45^{0}  sec^{2} 45^{0} cosec^{2} 45^{0} = 1 + 1  (√2)^{2} x (√2)^{2} = 2  4 = 2 26) If the value of tan^{2} θ + tan^{4} θ = 1, what will be the value of cos^{2} θ + cos^{4} θ?
Answer: (b) 1 Explanation: Given, tan^{2} θ + tan^{4} θ = 1 …. (i) From equation (i), tan^{2} θ ( 1 + tan^{2} θ ) = 1 tan^{2} θ ( sec^{2} θ ) = 1 [As according to the trigonometric identity, sec^{2} θ  tan^{2} θ = 1] tan^{2} θ = 1/ sec^{2} θ tan^{2} θ = cos^{2} θ ….(ii) Now, cos^{2} θ + cos^{4} θ = ? => cos^{2} θ + (cos^{2})^{2} θ => tan^{2} θ + (tan^{2})^{2} θ => tan^{2} θ + tan^{4} θ = 1 {from equation (i)} 27) If 4sin^{2} θ = 3, and θ is a positive acute angle, what is the value of tan θ  cot θ/2?
Answer: (b) 0 Explanation: Given, 4 sin^{2} θ = 3 Or, sin^{2} θ = 3/4 Or, sin θ = √3/2 So, θ = 60^{0} Now, tan θ  cot θ/2 = ? Put θ = 60^{0} = tan 60^{0}  cot 60^{0}/2 = tan 60^{0}  cot 30^{0} = √3  √3 = 0 28) If the value of sin A + cosec A = 2, then what is the value of sin7 A + cosec7 A?
Answer: (c) 2 Explanation: It is given that sin A + cosec A = 2 ……(i) On putting A = 90^{0}, then above condition will satisfy sin 90^{0} + cosec 90^{0} = 2 or, 1 + 1 = 2 (as the equation satisfies, so, A = 90^{0}) Now, sin^{7} A + cosec^{7} A = ? => sin^{7} 90^{0} + cosec^{7} 90^{0} => 1^{7} + 1^{7} = 2 29) What is the value of (2tan 30^{0}) / (1 + tan^{2} 30^{0})?
Answer: (c) sin 60^{0} Explanation: tan 30^{0} = 1/√3 (2tan 30^{0}) / (1 + tan^{2} 30^{0}) = ? = (2 x 1/√3) / (1 + (1/√3)^{2}) = (2/√3) / (4/3) = 6/4√3 = Or √3/2, which is equal to option C, i.e., sin60^{0}. 30) What is the value of (sin 30^{0} + cos 60^{0})  (sin 60^{0} + cos 30^{0})?
Answer: (c) 1 + √3 Explanation: Let's see the values  sin 30^{0} = 1/2 cos 60^{0} = 1/2 sin 60^{0} = √3/2 cos 30^{0} = √3/2 So, (1/2 + 1/2)  (√3/2 + √3/2) = 1  2√3/2 Or, 1  √3 31) If 1 + cos^{2} θ is equal to 3 sin θ.cos θ, then what is the value of cot θ?
Answer: (a) 1 Explanation: It is given that, 1 + cos^{2} θ = 3 cos θ.sin θ On dividing both sides by sin^{2} θ, we will get 1+cos^{2} θ / sin^{2} θ = 3 cos θ.sin θ/sin^{2} θ cosec^{2} θ + cot^{2} θ = 3 cot θ => 1 + cot^{2} θ + cot^{2} θ = 3 cot θ [because 1 + cot^{2} θ = cosec^{2} θ] => 1 + 2cot^{2} θ = 3 cot θ => 2 cot^{2} θ = 3 cot θ  1 Let's try to put θ = 45^{0} => 2cot^{2} 45^{0}  3 cot 45^{0} + 1 = 0 => 2 3 + 1 = 0 => 0 = 0 (satisfies) So, θ = 45^{0} cot θ = cot 45^{0} = 1 32) If the value of tan θ = 4/3, then which of the following is the correct value of (3 sin θ + 2 cos θ) / (3 sin θ  2 cos θ) =?
Answer: (d) 3 Explanation: It is given that, tan θ = 4/3 => sin θ / cos θ = 4/3 So sin θ = 4, and cos θ = 3 Now, on putting the values of sin θ and cos θ in (3 sin θ + 2 cos θ) / (3 sin θ  2 cos θ), we will get  = 3x4 + 2x3/ 3x4  2x3 = 18/6 = 3 33) If the value of tan 15^{0} is 2  √3, then what is the value of tan 15^{0} cot 75^{0} + tan 75^{0} cot 15^{0}?
Answer: (a) 14 Explanation: tan 15^{0} cot 75^{0} + tan 75^{0} cot 15^{0} = ? tan 15^{0} cot (90^{0}  15^{0}) + tan (90^{0}  15^{0}) cot 15^{0} [as cot (90^{0}  θ) = tan θ, and tan (90^{0}  θ) = cot θ] = tan^{2} 15^{0} + cot^{2} 15^{0} …..(i) cot 15^{0} = 1/tan 15^{0} = 1 / 2√3 = (1 / 2√3) x (2+√3 / 2+√3) So, cot 15^{0} = 2 + √3 So, on putting the values of cot 15^{0} and tan 15^{0} in equation (i), we will get = (2  √3)^{2} + (2 + √3)^{2} = 4 + 3  2√3 + 4 + 3 + 2√3 = 14 34) Which of the following is the correct relation between A and B, if A = tan 11^{0} . tan 29^{0}, and B = 2 cot 61^{0} . cot 79^{0}?
Answer: (d) 2A = B Explanation: Given A = tan 11^{0} . tan 29^{0}, and B = 2 cot 61^{0} . cot 79^{0} A / B = tan 11^{0} . tan 29^{0} / 2 cot 61^{0} . cot 79^{0} = [tan 11^{0} . tan 29^{0}] / [2 cot (90^{0}  29^{0}) . cot (90^{0}  11^{0})] = tan 11^{0} . tan 29^{0} / 2 tan 11^{0} . tan 29^{0} = 1/2 So, A/B = 1/2 Or, 2A = B 35) If sin θ x cos θ = 1/2, then what is the value of sin θ  cos θ?
Answer: (a) 0 Explanation: Given sin θ x cos θ = 1/2 On multiplying both sides by 2, we will get  2 sin θ x cos θ = 1 sin 2θ = 1 (because sin 2θ = 2 sinθ cosθ) So, 2θ = 90^{0} => θ = 45^{0} Therefore, sin θ  cos θ = ? => sin 45^{0}  cos 45^{0} = 1/√2  1/√2 = 0 36) If the value of sin(θ + 30^{0}) is 3/√12, then what is the value of cos^{2} θ?
Answer: (a) 3/4 Explanation: Given sin (θ + 30^{0}) = 3/√12 It can be written as sin (θ + 30^{0}) = 3/2√3 Or, sin (θ + 30^{0}) = √3/2 => sin (θ + 30^{0}) = sin 60^{0} => θ + 30^{0} = 60^{0} => θ = 30^{0} On putting θ = 30^{0}, in cos^{2} θ, we will get cos^{2} 30^{0} = (√3/2)^{2} = 3/4 37) If the value of 4 cos^{2}θ  4√3 cos θ + 3 = 0, then what is the value of θ?
Answer: (b) 300 Explanation: In this example, we can find the value of θ by putting values given in options. It is the hit and trial approach. The value that satisfies the given equation will be considered as the value of θ. Given 4 cos^{2}θ  4√3 cos θ + 3 = 0 So, in option A θ = 60^{0} is given, let's put it and see whether the equation will be satisfied or not  4 cos^{2}60^{0}  4√3 cos 60^{0} + 3 = 0 4 x (1/2)^{2}  4√3 x 1/2 + 3 = 0 => 4/4  4/2√3 + 3 = 0 => 4  4/2√3 = 0 => 4(1  1/2√3) = 0 (will not satisfy the equation) So, in option B, θ = 30^{0} is given, let's put it and see whether the equation will be satisfied or not  4 cos^{2}30^{0}  4√3 cos 30^{0} + 3 = 0 4 x (√3 /2)^{2}  4√3 x √3/2 + 3 = 0 => 3  6 + 3 = 0 => 6  6 = 0 => 0 = 0 (Equation satisfied) So, option B is correct, and the value of θ is 30^{0}. 38) Which of the following is the correct value of cos^{2} 55^{0} + cos^{2} 35^{0} + sin^{2} 65^{0} + sin^{2} 25^{0}?
Answer: (c) 2 Explanation: cos^{2} 55^{0} + cos^{2} 35^{0} + sin^{2} 65^{0} + sin^{2} 25^{0} => cos^{2} (90^{0}  35^{0}) + cos^{2} 35^{0} + sin^{2} 65^{0} + sin^{2} (90^{0}  65^{0}) => (sin^{2} 35^{0} + cos^{2} 35^{0}) + (sin^{2} 65^{0} + cos^{2} 65^{0}) => 1 + 1 = 2 39) If the value of tan 9^{0} = p/q, then what is the value of sec^{2} 81^{0}/ 1 + cot^{2} 81^{0}?
Answer: (c) q^{2}/p^{2} Explanation: sec^{2} 81^{0}/ 1 + cot^{2} 81^{0} = sec^{2} 81^{0}/ cosec^{2} 81^{0} = (1/cos^{2} 81^{0}) / (1/sin^{2} 81^{0}) = sin^{2} 81^{0} / cos^{2} 81^{0} = tan^{2} 81^{0} = tan^{2} (90^{0}  9^{0}) = cot^{2} 9^{0} = q^{2} / p^{2} 40) If cot 45^{0}.sec 60^{0} = A tan 30^{0}.sin 60^{0}, then which of the following is the correct value of A?
Answer: (a) 4 Explanation: Given cot 45^{0}.sec 60^{0} = A tan 30^{0}.sin 60^{0} So, 1 x 2 = A 1/√3 x √3/2 => 2 = A/2 So, A = 4 41) If the value of sec^{2} θ + tan^{2} θ = 7, then what is the value of θ?
Answer: (c) 60^{0} Explanation: Given sec^{2} θ + tan^{2} θ = 7 => 1 + tan^{2} θ + tan^{2} θ = 7 => 2tan^{2} θ + 1 = 7 => 2tan^{2} θ = 6 => tan^{2} θ = 3 => tan θ = √3 Or θ = 60^{0} We can also solve this question by using the hit and trial approach. We can directly check the values of θ given in options. The value that will satisfy the given condition will be the value of θ. 42) Which of the following is the correct value of (3 / 1+tan^{2} θ) + 2 sin^{2} θ + (1 / 1+cot^{2} θ)?
Answer: (a) 3 Explanation: (3 / 1+tan^{2} θ) + 2 sin^{2} θ + (1 / 1+cot^{2} θ) = ? According to the trigonometric identities, the given equation can be written as  = 3/sec^{2} θ + 2 sin^{2} θ + 1/cosec^{2} θ = 3cos^{2} θ + 2 sin^{2} θ + sin^{2} θ = 3cos^{2} θ + 3sin^{2} θ = 3(cos^{2} θ + sin^{2} θ) = 3 43) If the value of tan^{2} A = 1 + 2tan^{2} B, then what is the value of √2 cosA  cosB?
Answer: (a) 0 Explanation: Given tan^{2} A = 1 + 2tan^{2} B => sec^{2} A  1 = 1 + 2 (sec^{2} B  1) => sec^{2} A  1 = 1 + 2 sec^{2} B  2 => sec^{2} A  1 = 2 sec^{2} B  1 => 1/cos^{2} A = 2/cos^{2} B => cos^{2} B = 2cos^{2} A => or, cos B = √2 cos A => So, √2 cos A  cos B = 0 44) What will be the numerical value of (4 sec^{2} 30^{0} + cos^{2} 60^{0}  tan^{2} 45^{0}) / (sin^{2} 30^{0} + cos^{2} 30^{0})?
Answer: (a) 55/12 Explanation: Given: (4 sec^{2} 30^{0} + cos^{2} 60^{0}  tan^{2} 45^{0}) / (sin^{2} 30^{0} + cos^{2} 30^{0}) We have to put the numerical values, = [4 (2/√3)^{2} + (½)^{2}  (1)^{2}] / 1 => sec^{2} A  1 = 1 + 2 (sec^{2} B  1) => sec^{2} A  1 = 1 + 2 sec^{2} B  2 => sec^{2} A  1 = 2 sec^{2} B  1 => 1/cos^{2} A = 2/cos^{2} B => cos^{2} B = 2cos^{2} A => or, cos B = √2 cos A => So, √2 cos A  cos B = 0 45) If the value of tan A = 2, then what is the value of (cosec^{2} A  sec^{2} A) / (cosec^{2} A + sec^{2} A)?
Answer: (d)  3/5 Explanation: Given: tan A = 2 (cosec^{2} A  sec^{2} A) / (cosec^{2} A + sec^{2} A) = ? On dividing above equation with cosec^{2} A, we will get  = (1  tan^{2} A) / (1 + tan^{2} A) = (1  2^{2}) / (1 + 2^{2}) [because tan A = 2] =  3/5 46) Which of the following is the correct value of (5/sec^{2} θ) + 3 sin^{2} θ + (2 / 1+cot^{2} θ)?
Answer: (c) 5 Explanation: (5 / sec^{2} θ) + 3 sin^{2} θ + (2 / 1+cot^{2} θ) = ? According to the trigonometric identities, the given equation can be written as  = 5cos^{2} θ + 3 sin^{2} θ + 2/cosec^{2} θ = 5cos^{2} θ + 3 sin^{2} θ + 2sin^{2} θ = 5cos^{2} θ + 5sin^{2} θ = 5(cos^{2} θ + sin^{2} θ) = 5 47) Which of the following is the correct numerical value of 5 tan^{2} A  5 sec^{2} A + 1?
Answer: (d) 4 Explanation: 5 tan^{2} A  5 sec^{2} A + 1 = ? = 5 (tan^{2} A  sec^{2} A) + 1 = 5 ((sin^{2} A/cos^{2} A)  (1/cos^{2} A)) + 1 = 5((sin^{2} A  1) / cos^{2} A) + 1 = 5( cos^{2} A / cos^{2} A) + 1 =  5 + 1 =  4 48) The value of cot 30^{0}/tan 60^{0} is 
Answer: (c) 1 Explanation: tan 60^{0} = √3, cot 30^{0} = √3 So, cot 30^{0}/tan 60^{0} = √3 / √3 = 1 49) If the value of tanP + secP = a, then what is the value of cosP?
Answer: (a) 2a/a^{2} + 1 Explanation: It is given that, tanP + secP = a ……(i) As we know, the trigonometric identity, sec^{2} P  tan^{2} P = 1 {we assume θ = P} So, we can apply the formula a^{2}  b^{2} = (a  b) (a + b) => (sec P  tan P) (sec P + tan P) = 1 => (sec P  tan P) x a = 1 => sec P  tan P = 1/a …..(ii) So, from equation (i) and (ii), we will get  2sec P = a + 1/a sec P = a^{2}+1 / 2a So, cos P = 2a / a^{2}+1 [as sec P = 1/cosP] 50) Suppose cos θ + sin θ = √2 cos θ, then which of the following is the correct value of cos θ  sin θ?
Answer: (b) √2 sin θ Explanation: It is given that, cos θ + sin θ = √2 cos θ …..(i) On squaring both sides, we will get, (cos θ + sin θ)^{2} = (√2 cos θ)^{2} => cos^{2} θ + sin^{2} θ + 2 sin θ cos θ = 2 cos^{2} θ Or, 2cos^{2} θ  cos^{2} θ  sin^{2} θ = 2 sinθ cosθ => cos^{2} θ  sin^{2} θ = 2 sin θ cos θ => (cos θ + sin θ) (cos θ  sin θ) = 2 sin θ cos θ => (√2 cos θ) (cos θ  sin θ) = 2 sin θ cos θ [from equation (i)] => (cos θ  sin θ) = 2 sinθ cosθ / √2 cos θ = √2 sin θ
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